Methods for Building and Validating Equations for Physiology-Based Mathematical Models: Glucose Metabolism and Type 2 Diabetes in the Archimedes Model.

OBJECTIVES: Describe steps for deriving and validating equations for physiology processes for use in mathematical models. Illustrate the steps using glucose metabolism and Type 2 diabetes in the Archimedes model. METHODS AND RESULTS: The steps are as follows: identify relevant variables, describe their relationships, identify data sources that relate the variables, correct for biases in data sources, use curve fitting algorithms to estimate equations, validate the accuracy of curve fitting against empirical data, perform partially and fully independent external validations, examine any discrepancies to determine causes and make corrections, and periodically update and revalidate equations as necessary. Specific methods depend on the available data. Specific data sources and methods are illustrated for equations that represent the cause of Type 2 diabetes and its effect on fasting plasma glucose in the Archimedes model. Methods for validating the equations are illustrated. Applications enabled by including physiological equations in healthcare models are discussed. CONCLUSIONS: The methods can be used to derive equations that represent the relationships between physiological variables and the causes of diseases and that validate well against empirical data.

Diabetes Risk Calculator: a simple tool for detecting undiagnosed diabetes and pre-diabetes.

OBJECTIVE: The objective of this study was to develop a simple tool for the U.S. population to calculate the probability that an individual has either undiagnosed diabetes or pre-diabetes. RESEARCH DESIGN AND METHODS: We used data from the Third National Health and Nutrition Examination Survey (NHANES) and two methods (logistic regression and classification tree analysis) to build two models. We selected the classification tree model on the basis of its equivalent accuracy but greater ease of use. RESULTS: The resulting tool, called the Diabetes Risk Calculator, includes questions on age, waist circumference, gestational diabetes, height, race/ethnicity, hypertension, family history, and exercise. Each terminal node specifies an individual's probability of pre-diabetes or of undiagnosed diabetes. Terminal nodes can also be used categorically to designate an individual as having a high risk for 1) undiagnosed diabetes or pre-diabetes, 2) pre-diabetes, or 3) neither undiagnosed diabetes or pre-diabetes. With these classifications, the sensitivity, specificity, positive and negative predictive values, and receiver operating characteristic area for detecting undiagnosed diabetes are 88%, 75%, 14%, 99.3%, and 0.85, respectively. For pre-diabetes or undiagnosed diabetes, the results are 75%, 65%, 49%, 85%, and 0.75, respectively. We validated the tool using v-fold cross-validation and performed an independent validation against NHANES 1999-2004 data. CONCLUSIONS: The Diabetes Risk Calculator is the only currently available noninvasive screening tool designed and validated to detect both pre-diabetes and undiagnosed diabetes in the U.S. population.

Clinical outcomes and cost-effectiveness of strategies for managing people at high risk for diabetes.

BACKGROUND: Lifestyle modification can forestall diabetes in high-risk people, but the long-term cost-effectiveness is uncertain. OBJECTIVE: To estimate the effects of the lifestyle modification program used in the Diabetes Prevention Program (DPP) on health and economic outcomes. DESIGN: Cost-effectiveness analysis using the Archimedes model. DATA SOURCES: Published basic and epidemiologic studies, clinical trials, and Kaiser Permanente administrative data. TARGET POPULATION: Adults at high risk for diabetes (body mass index >24 kg/m2, fasting plasma glucose level of 5.2725 to 6.9375 mmol/L [95 to 125 mg/dL], 2-hour glucose tolerance test result of 7.77 to 11.0445 mmol/L [140 to 199 mg/dL]). TIME HORIZON: 5 to 30 years. PERSPECTIVE: Patient, health plan, and societal. INTERVENTIONS: No prevention, DPP's lifestyle modification program, lifestyle modification begun after a person develops diabetes, and metformin. MEASUREMENTS: Diagnosis and complications of diabetes. RESULTS OF BASE-CASE ANALYSIS: Compared with no prevention program, the DPP lifestyle program would reduce a high-risk person's 30-year chances of getting diabetes from about 72% to 61%, the chances of a serious complication from about 38% to 30%, and the chances of dying of a complication of diabetes from about 13.5% to 11.2%. Metformin would deliver about one third the long-term health benefits achievable by immediate lifestyle modification. Compared with not implementing any prevention program, the expected 30-year cost/quality-adjusted life-year (QALY) of the DPP lifestyle intervention from the health plan's perspective would be about 143,000 dollars. From a societal perspective, the cost/QALY of the lifestyle intervention compared with doing nothing would be about 62,600 dollars. Either using metformin or delaying the lifestyle intervention until after a person develops diabetes would be more cost-effective, costing about 35,400 dollars or 24,500 dollars per QALY gained, respectively, compared with no program. Compared with delaying the lifestyle program until after diabetes is diagnosed, the marginal cost-effectiveness of beginning the DPP lifestyle program immediately would be about 201,800 dollars. RESULTS OF SENSITIVITY ANALYSIS: Variability and uncertainty deriving from the structure of the model were tested by comparing the model's results with the results of real clinical trials of diabetes and its complications. The most critical element of uncertainty is the effectiveness of the lifestyle program, as expressed by the 95% CI of the DPP study. The most important potentially controllable factor is the cost of the lifestyle program. Compared with no program, lifestyle modification for high-risk people can be made cost-saving over 30 years if the annual cost of the intervention can be reduced to about 100 dollars. LIMITATIONS: Results depend on the accuracy of the model. CONCLUSIONS: Lifestyle modification is likely to have important effects on the morbidity and mortality of diabetes and should be recommended to all high-risk people. The program used in the DPP study may be too expensive for health plans or a national program to implement. Less expensive methods are needed to achieve the degree of weight loss seen in the DPP.

Archimedes: a trial-validated model of diabetes.

OBJECTIVE: To build a mathematical model of the anatomy, pathophysiology, tests, treatments, and outcomes pertaining to diabetes that could be applied to a wide variety of clinical and administrative problems and that could be validated. RESEARCH DESIGN AND METHODS: We used an object-oriented approach, differential equations, and a construct we call "features." The level of detail and realism was determined by what clinicians considered important, by the need to distinguish clinically relevant variables, and by the level of detail used in the conduct of clinical trials. RESULTS: The model includes the pertinent organ systems, more than 50 continuously interacting biological variables, and the major symptoms, tests, treatments, and outcomes. The level of detail corresponds to that found in general medical textbooks, patient charts, clinical practice guidelines, and designs of clinical trials. The model is continuous in time and represents biological variables continuously. As demonstrated in a companion article, the equations can simulate a variety of clinical trials and reproduce their results with good accuracy. CONCLUSIONS: It is possible to build a mathematical model that replicates the pathophysiology of diabetes at a high level of biological and clinical detail and that can be tested by simulating clinical trials.

Validation of the archimedes diabetes model.

OBJECTIVE: To validate the Archimedes model of diabetes and its complications for a variety of populations, organ systems, treatments, and outcomes. RESEARCH DESIGN AND METHODS: We simulated a variety of randomized controlled trials by repeating in the model the steps taken for the real trials and comparing the results calculated by the model with the results of the trial. Eighteen trials were chosen by an independent advisory committee. Half the trials had been used to help build the model ("internal" or "dependent" validations); the other half had not. Those trials comprise "external" or "independent" validations. RESULTS: A total of 74 validation exercises were conducted involving different treatments and outcomes in the 18 trials. For 71 of the 74 exercises there were no statistically significant differences between the results calculated by the model and the results observed in the trial. Considering only the trials that were never used to help build the model-the independent or external validations-the correlation was r = 0.99. Including all of the exercises, the correlation between the outcomes calculated by the model and the outcomes seen in the trials was r = 0.99. When the absolute differences in outcomes between the control and treatment groups were compared, the correlation coefficient was r = 0.97. CONCLUSIONS: The Archimedes diabetes model is a realistic representation of the anatomy, pathophysiology, treatments, and outcomes pertinent to diabetes and its complications for applications that involve the populations, treatments, outcomes, and health care settings spanned by the trials.

Archimedes: a new model for simulating health care systems--the mathematical formulation.

This paper designs an object-oriented, continuous-time, full simulation model for addressing a wide range of clinical, procedural, administrative, and financial decisions in health care at a high level of biological, clinical, and administrative detail. The full model has two main parts, which with some simplification can be designated "physiology models" and "models of care processes." The models of care processes, although highly detailed, are mathematically straightforward. However, the mathematics that describes human biology, diseases, and the effects of interventions are more difficult. This paper describes the mathematical formulation and methods for deriving equations, for a variety of different sources of data. Although Archimedes was originally designed for health care applications, the formulation, and equations are general and can be applied to many natural systems.